Lemma: An admissible rank-one difference

The matrix

is an admissible smooth ordering difference in XII-A's class.

Proof. It is smooth and symmetric under simultaneous exchange of species and spacetime slots. Every nonzero species is even and has ghost number zero. Since $u$ satisfies the complete shell equations, the matrix satisfies those equations in each slot, including the auxiliary relation $u_b=D{u}_{\gamma }$. Its graded antisymmetric part is zero, so $H+d$ has the same causal part as $H$. These are precisely the required conditions. No positivity or compact support condition on $d$ is imposed XIIA.